Fitting QENS data

Fitting QENS data#

Previously, some quasi-elastic neutron scattering (QENS) data has been simulated and reduced and can now be analysed with easyCore. Before the analysis can begin, it is necessary to load the experimental data and check that it looks reasonable. The data can be loaded with np.loadtxt as the data has been stored in a simple space-separated column file.

import numpy as np

qens_data = np.loadtxt('../4-reduction/qens_energy_transfer_unknown_quasi_elastic_3_pulse.dat', unpack=True)

---------------------------------------------------------------------------
FileNotFoundError                         Traceback (most recent call last)
Cell In[1], line 3
      1 import numpy as np
----> 3 qens_data = np.loadtxt('../4-reduction/qens_energy_transfer_unknown_quasi_elastic_3_pulse.dat', unpack=True)

File /opt/hostedtoolcache/Python/3.10.13/x64/lib/python3.10/site-packages/numpy/lib/npyio.py:1373, in loadtxt(fname, dtype, comments, delimiter, converters, skiprows, usecols, unpack, ndmin, encoding, max_rows, quotechar, like)
   1370 if isinstance(delimiter, bytes):
   1371     delimiter = delimiter.decode('latin1')
-> 1373 arr = _read(fname, dtype=dtype, comment=comment, delimiter=delimiter,
   1374             converters=converters, skiplines=skiprows, usecols=usecols,
   1375             unpack=unpack, ndmin=ndmin, encoding=encoding,
   1376             max_rows=max_rows, quote=quotechar)
   1378 return arr

File /opt/hostedtoolcache/Python/3.10.13/x64/lib/python3.10/site-packages/numpy/lib/npyio.py:992, in _read(fname, delimiter, comment, quote, imaginary_unit, usecols, skiplines, max_rows, converters, ndmin, unpack, dtype, encoding)
    990     fname = os.fspath(fname)
    991 if isinstance(fname, str):
--> 992     fh = np.lib._datasource.open(fname, 'rt', encoding=encoding)
    993     if encoding is None:
    994         encoding = getattr(fh, 'encoding', 'latin1')

File /opt/hostedtoolcache/Python/3.10.13/x64/lib/python3.10/site-packages/numpy/lib/_datasource.py:193, in open(path, mode, destpath, encoding, newline)
    156 """
    157 Open `path` with `mode` and return the file object.
    158 
   (...)
    189 
    190 """
    192 ds = DataSource(destpath)
--> 193 return ds.open(path, mode, encoding=encoding, newline=newline)

File /opt/hostedtoolcache/Python/3.10.13/x64/lib/python3.10/site-packages/numpy/lib/_datasource.py:533, in DataSource.open(self, path, mode, encoding, newline)
    530     return _file_openers[ext](found, mode=mode,
    531                               encoding=encoding, newline=newline)
    532 else:
--> 533     raise FileNotFoundError(f"{path} not found.")

FileNotFoundError: ../4-reduction/qens_energy_transfer_unknown_quasi_elastic_3_pulse.dat not found.

With the data read in, we can produce a quick plot.

import matplotlib.pyplot as plt

plt.errorbar(*qens_data)
plt.xlabel('$\omega$/meV')
plt.ylabel('$I(\omega)$')
plt.show()

The QENS data covers a slightly broader range than we want to investigate. Therefore, we will crop the data so that we are only using points between -0.06 and 0.06 meV.

q, i, di = qens_data[:, np.where((qens_data[0] > -0.06) & (qens_data[0] < 0.06))[0]]

We now want to consider the mathematical model to be used in the analysis. QENS data is typically analysed with by fitting a Lorentzian function to the data, which has the functional form

(5)#\[ I(\omega) = \frac{A\tau}{\pi\big[(\omega - \omega_0) ^ 2 + \tau ^ 2\big]}, \]

where \(A\) is a scale factor, \(\tau\) is the half-width at half-maximum, and \(\omega_0\) is the centre offset.

Task

Write a function that implements Eqn. (5), called lorentzian.

Click below to show code solution

The Lorentzian function is then typically convolved with a Gaussian distribution that is centred at zero with some known width, \(\sigma\). For this project, we will model \(\sigma\).

from scipy.stats import norm

def intensity(omega):
    """
    The convolution of a Gaussian and the Lorentzian.
    
    :omega: the energy transfer values to calculate over.

    :return: intensity values.
    """
    gauss = norm(0, sigma.raw_value).pdf(omega)
    gauss /= gauss.sum()
    return np.convolve(lorentzian(omega), gauss, 'same')

This means that there are a total of four parameters in our model.

Task

Create four Parameter objects, for \(A\), \(\tau\), \(\omega_0\), and \(\sigma\). Each should have an initial value and a uniform prior distribution, based on the values given in Table 2.

Table 2 Parameter values for the spherical model.#
Parameter	Initial Value	Min	Max
\(A\)	10	1	100
\(\tau\)	8.0 × 10^-3	1.0 × 10^-4	1.0 × 10^-2
\(\omega_0\)	1.0 × 10^-3	0	2.0 × 10^-3
\(\sigma\)	1.0 × 10^-3	1.0 × 10^-5	1.0 × 10^-1

Click below to show code solution

It is now possible to compare our model at the initial estimates to the simulated data.

plt.errorbar(q, i, di, marker='.', ls='', color='k')
plt.plot(q, intensity(q), '-')
plt.xlabel('$\omega$/meV')
plt.ylabel('$I(\omega)$')
plt.show()

Task

Using easyCore obtain maximum likelihood estimates for the four parameters of the model from comparison with the data.

Click below to show code solution

We can then plot the model and the data together as before and print the values of the parameters along with their uncertainties.

plt.errorbar(q, i, di, marker='.', ls='', color='k')
plt.plot(q, intensity(q), '-')
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.show()

A.value

tau.value

omega_0.value

sigma.value